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Help needed! What have I done wrong here?

Given the metric $$ds^2 = dr^2+r^2d\theta^2$$

And $$R_{ij}=\nabla_i \nabla_j f(r)$$ where $\nabla_i$ is a covariant derivative, and $f=f(r)$ is a scalar function.

I wish to show that $f'(r)\over r$ is a constant. However, I can't see how.

What I have thought about:

$f$ being a scalar function means that the covariant derivatives reduce to $\partial_i, \,\,\partial_j$

This metric describes a flat space. Therefore the Ricci scalar $R=g^{ij}R_{ij}=0$. So $$g^{ij}\partial_i\partial_j f(r)=0$$ For this metric, it is equivalent to $$g^{rr}\partial_r\partial_r f(r)=0$$ which is equivalent to $${\partial^2\over \partial r^2}f=0$$

Unfortunately here is where I've got stuck. How does the constant condition follow?

Thank you.

Harrold
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  • $R_{ij} = 0$ doesn't imply $\Gamma_{pq}^{r} = 0$ in a coordinate system where $g_{ij}$ isn't a constant. i.e. your $\nabla_{i} \ne \partial_i$ in polar coordinates. – achille hui Feb 21 '13 at 16:11

0 Answers0